Judge Swain has handled prominent cases in recent years. She was the presiding judge who accepted the guilty plea of SAC Capital Advisors, the giant group of hedge funds that the billionaire investor Steven A. Cohen led. And she presided over the lengthy criminal trial of several former employees of Bernard Madoff, who ran a huge Ponzi scheme.

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Puerto Rico's $123 billion in debt and pension obligations far exceeds the $18 billion bankruptcy filed by Detroit in 2013. Some of the island's debt of $74 billion is held by hedge funds.

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The lawsuit seeking bankruptcy-like relief, filed by a representative of Puerto Rico, lists Cede & Company of New York as the largest unsecured creditor, with a claim of over $12 billion. The filing described Cede as a bond trustee that represents "publicly held beneficial holders of the bond debt."

Judge Swain has the option of holding proceedings in the case either in her courtroom in Manhattan or in federal court in San Juan, said Martin J. Bienenstock, the head of the bankruptcy practice at the New York law firm Proskauer Rose and an outside counsel for the Financial Oversight and Management Board for Puerto Rico.

Because Puerto Rico's case will be the first to be heard under a federal law for insolvent territories, called Promesa, Judge Swain will be entering uncharted waters.

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Judge Swain displayed her deliberative nature when she declined to immediately accept the guilty plea from Mr. Cohen's former hedge fund in November 2013. Lawyers for SAC appeared in court before her to enter the plea, but she said she wanted to take several months to review the matter -- which included the payment of a $1.2 billion fine.

Then, in April 2014, when Judge Swain held a hearing to formally accept SAC's plea to insider-trading charges, she took a swipe at the firm's top management. At the hearing, she said the leadership of the firm had "ignored blatant red flags" and "rewarded this illegal behavior."

She concluded her comments saying she hoped that Mr. Cohen, who was not charged with any wrongdoing by prosecutors, would run his new firm, Point72 Asset Management, with "respect and compliance with the law."